Math 2111-Matrix Algebra and Applications (Written by Dr.
Hon-Ming HO)
Lecture Notes 9: Compositions of Linear Transformations, Transposes of Matrices
Composition Theorem (Composition of two linear transformations is again a linear transformation):
Given two linear transformations and with standard matrices and respectively. Then the
composition is a linear transformation with standard matrix (the product matrix of and ).
Symbolically, we have
Proof:
Suppose that and are linear transformations with standard matrices and respectively. To
show that is linear, we let and be two testing vectors in and and two scalars. Define
. Our goal is to show that .
The proof that the composition is linear is complete. So by the standard matrix theorem for linear transformation, the
linear transformation has a unique standard matrix satisfying the following condition: for each column vector in
, we have We are going to prove that the new single matrix above is exactly equal to the product
matrix . What are the relation between elements of and elements of and ? We will show that
for each column vector in . If we succeed to do so, then by the uniqueness property, we must have .
Lecture Notes 9 (written by Dr. Hon-Ming HO)-page 1
Example 1:
Prove that by using the theory of matrices and linear transformations.
Solution to Example 1:
We define a mapping as follows: for any given column vector in , is defined as the column vector
obtained by rotating in anticlockwise direction about the origin through angle . Then can be regarded as the
column vector obtained by two kinds of operations: the first one is to rotate in anticlockwise direction about the origin
through angle and then rotate in anticlockwise direction about the origin through angle to obtain .
Since and are linear transformations, their composition is also linear. Thus by the composition theorem above, we have
! "
Thus we conclude that .
Theorem 1:
Let be an matrix and an matrix. If are column vectors of the matrix , then we have
Proof:
There are more than one method to prove the matrix identity above. Since we would like to emphasize the relation between
linear transformations and matrices, we prove the identity above from transformational point of view as follows.
The underlying senses of and :
Original Prove that . 1) Object is a rectangular array of numbers
Problem : having rows and columns.
4
2) Object is another rectangular array of
1 3 numbers having rows and columns.
2 How do we enter into transformational point of view? We shift our focus of attention about from its underlying sense to its second sense.
Let matrix . We define as Transformational senses of and :
1) We regard as the unique standard matrix of
follows: for all in , let . Prove that . Once we
a linear transformation with
prove it, we can conclude that and have the same standard matrix
Transformational for each in .
which implies that by the composition theorem o page 1.
Problem : 2) We regard as the unique standard matrix of
another linear transformation
Before we enter into the proof, let us recall that each linear system has four
with for each in .
equivalent forms: they are matrix equation form, vector equation form,
augmented matrix form and the original system (see Lecture Notes 4, page 5). Let
. Then
Since is the standard matrix of the composition by the composition
theorem, we conclude that .
Lecture Notes 9 (written by Dr. Hon-Ming HO)-page 2
Algebraic Properties of Matrices:
1) Let be an matrix, and two matrices. Then we have .
2) Let and be two matrices, an matrix. Then we have .
3) Let be an matrix, an matrix. For any real number , we have .
4) Let be an matrix. Let be the identity matrix. Then .
5) Let be an matrix, an matrix, an matrix. Then we have .
Definition (Transpose):
The transpose of an matrix is defined to be the matrix, denoted by , whose -entry is the -entry
of . That is,
Remark: The -entry of means the entry of lying in the row and column.
Theorem 2:
Let be an matrix, an matrix. Then we have .
Proof:
To show that , it is sufficient for us to show that -entry of is equal to -entry of .
Lecture Notes 9 (written by Dr. Hon-Ming HO)-page 3
